How fragile is your network? More than you think.

The paper defines fragility F_δ(G) via the complete-graph baseline f_comp and claims that complete equitable bipartite graphs (CEB_n) are asymptotically robust (F_δ(CEB_n) → 0) . It asserts an exact formula for r* on CEB graphs by “following the logic” used for complete graphs (its Eq. (20)), but does not supply a rigorous bipartite analogue of the ‘splitting squares’ extremal step or the discrete convexity argument needed to justify the maximization of kept edges under an LCC ≤ c constraint . The subsequent limit analysis for F_δ treats several special cases (e.g., c = n − k, k fixed; and δ = 1/2) and then generalizes with heuristic language rather than a uniform argument for all fixed δ ∈ (0,1) . By contrast, the model’s solution provides the needed combinatorial extremal argument on K_{p,q}, uses the discrete convexity of S(s) = ⌊s^2/4⌋, and constructs near-tight subgraphs to show f_CEB_n(δ) = f_comp(δ) + O(1/n), hence F_δ(CEB_n) = O(1/n) → 0. The model is correct; the paper’s conclusion is plausible and matches the model, but the proof is incomplete at key steps.